Gas Laws: Boyle's, Charles's and Avogadro's Law

• Gases are compressible and expandable, completely fill containers, and have lower densities than liquids and solids.

Gas Pressure

• Pressure is the force per unit area exerted by gas molecules on container walls.
  • $Pressure = \frac{Force}{Area}$
• Pascal (Pa): the SI unit of pressure, $1 Pa = 1 \frac{N}{m^2}$.
• Other pressure units:
  • 1 atm = 760 mmHg = 760 torr = 101.325 kPa
  • 1 bar = $10^5$ Pa

The Gas Laws

• These laws describe relationships between gas volume (V), pressure (P), temperature (T), and the number of moles (n).

Boyle's Law

• At constant temperature and number of moles, gas volume is inversely proportional to pressure: $V \propto \frac{1}{P}$.
  • $P_1V_1 = P_2V_2$

Charles's Law

• At constant pressure and number of moles, gas volume is directly proportional to absolute temperature: $V \propto T$.
  • $\frac{V_1}{T_1} = \frac{V_2}{T_2}$
• Use Kelvin for temperature: $T(K) = t(°C) + 273.15$

Avogadro's Law

• At constant pressure and temperature, gas volume is directly proportional to the number of moles: $V \propto n$.
  • $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

The Ideal Gas Law

Ideal Gas Law

• The ideal gas law combines Boyle's, Charles', and Avogadro's laws: $PV = nRT$.
• The ideal gas constant is $R = 0.08206 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$.
• Standard Temperature and Pressure (STP) is $0°C$ (273.15 K) and 1 atm.
• At STP, one mole of an ideal gas occupies 22.4 L, which is the standard molar volume.

Gas Density and Molar Mass

• Density is $d = \frac{m}{V}$
• Molar mass is $M = \frac{m}{n}$
• Use the ideal gas law to find density: $d = \frac{PM}{RT}$
• Use the ideal gas law to find molar mass: $M = \frac{dRT}{P}$ or $M = \frac{mRT}{PV}$

Gas Mixtures and Partial Pressures

Dalton's Law of Partial Pressures

• The total pressure of a gas mixture equals the sum of each gas's partial pressures: $P_{total} = P_1 + P_2 + P_3 +...$
  • $P_i = n_i \frac{RT}{V}$

Mole Fraction

• The mole fraction of a gas in a mixture is the ratio of its moles to the total moles: $X_i = \frac{n_i}{n_{total}}$
• A gas's partial pressure equals the total pressure times its mole fraction: $P_i = X_i \cdot P_{total}$

The Kinetic Molecular Theory of Gases

Main Assumptions

• Gas particle size is negligible compared to the space between them.
• Gas particles move randomly and continuously.
• Collisions between particles and container walls are elastic (kinetic energy is conserved).
• There are no intermolecular forces between gas particles.
• The average kinetic energy of gas particles is proportional to absolute temperature.

Kinetic Energy and Root Mean Square Speed

• Average kinetic energy is $KE_{avg} = \frac{3}{2}RT$
• Root mean square speed is $u_{rms} = \sqrt{\frac{3RT}{M}}$, where M is molar mass in kg/mol.

Graham's Law of Effusion

• The rate of effusion is inversely proportional to the square root of molar mass: $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$
• Effusion: gas escaping through a small hole.

Real Gases: Deviations from Ideal Behavior

Real Gases

• Real gases deviate from ideal behavior at high pressure and low temperature.
  • Intermolecular forces exist between gas particles.
  • Gas particles have a finite volume.

van der Waals Equation

• The van der Waals equation corrects for deviations from ideal behavior: $[P + a(\frac{n}{V})^2](V - nb) = nRT$
  • $a$ accounts for intermolecular forces.
  • $b$ accounts for the finite volume of gas particles.
  • $a$ and $b$ are van der Waals constants specific to each gas.